Finding the external fields $h_{i}$ and the coupling parameters $J_{ij}$ can fit the same means and pairwise covariance as the experiment data, using the equations

$\left<s_{i}\right>_{Ising} = \left<s_{i}\right>_{data} $

$\left<s_{i}s_{j}\right>_{Ising} = \left<s_{i}s_{j}\right>_{data} $.

We can solve the equations by iterations $h_{i}=h_{i}+\delta h_{i}$ and $J_{ij}=J_{ij}+\delta J_{ij}$ then make changes

$\delta h_{i}=\eta \{\left<s_{i}\right>_{data} - \left<s_{i}\right>_{Ising} \} $

$\delta J_{ij}=\eta \{\left<s_{i}s_{j}\right>_{data} - \left<s_{i}s_{j}\right>_{Ising} \} $.

We use the data ACV_slice5 (6 neurons) as our sample and set $\eta = 0.1$. The cost function is

${(\left<s_{i}\right>_{data} - \left<s_{i}\right>_{Ising})}^{2} \leq 0.00000001$ && ${(\left<s_{i}s_{j}\right>_{data} - \left<s_{i}s_{j}\right>_{Ising})}^{2} \leq 0.00000001$.

The experiment result and simulation result see figure.

If the number of neurons is larger than 30, it is very slow to find the suitable $h_{i}$ and $J_{ij}$ for the good fitting.

The coefficient of $R^{2}$ is:

$R^{2}=1-\frac{\sum_{ij} \left(J_{ij}^{\text{approx}}-J_{ij}^{\text{Exact}} \right)^{2} }{\sum_{ij} \left(J_{ij}^{\text{Exact}}-J_{ij}^{\text{$\overline{Exact}$}} \right)^{2}}$ , with

$\overline{J_{ij}^{\text{Exact}}}=\frac {\sum_{i \neq j} J_{ij}^{\text{Exact}}}{N(N-1)}$.

The rms error is: $\sqrt{\frac{1}{N(N-1)} \sum_{i \neq j} (J_{ij}^\text{approx}-J_{ij}^\text{Exact})^{2} }$.

We set the mean equal to zero and the variance of normal distribution of h and J from $\frac{1}{\sqrt{200}}$: $\frac{1}{\sqrt{150}}$: $\frac{1}{\sqrt{100}}$:$\frac{1}{\sqrt{50}}$:$\frac{1}{\sqrt{20}} \approx 0.07 : 0.08 : 0.1 : 0.14 : 0.22 $.

We use the normal distribution of h and J to get the coefficients of $R^{2}$ and the RMSerror at T=1.